MGMT 355, Spring 2012

Midterm 1 (Take home)

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1)

Name:

2)

Name:

3)

Name

Section sits at 11:30AM

1:00 PM

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All group members must sign stating that you ALL worked together in completing this

examination

(1)

In the matter of constructing control charts the following information is available

about some measurable quantity on a product (the units are omitted):

Sample # obs. 1

obs. 2

obs. 3

obs. 4

1

5

4

3

8

2

4

7

7

9

3

6

2

4

8

4

3

4

5

6

(a) Obtain the control limits for a three-sigma . (7.5 points)

(b) Similarly, obtain the control limits for a three-sigma R-chart. (7.5 points)

(c) Using a rough graph show how you would use these charts (you can draw one rough

graph for any one of these). (5 points)

(d) Suppose that you ask your machine operator to stop everytime the control limits of

your R-chart is violated. What fraction of the times would you find the machine to

have been stopped when actually nothing was wrong? (5 points)

(e) Contruct the control limits for a p-chart so that your charts have a failure-rate or falsealarm rate of 5%. Assume that using samples of size 64 you found to be 0.05. (10

points)

(f) Suppose you want to construct a control chart that will have 97.5% chance of

including sample means. What should be the UCL and LCL of the x-bar chart when

the average of the sample mean of the associated product is 36 oz and standard

deviation of sample means is 0.15 oz. (Use Excel to get an accurate value)

(2)

Joseph King has ambitions to be mayor of Williston, North Dakota. Joe has determined the

breakdown of the steps to the nomination and has estimated normal and crash costs and times

for the campaign as follows (times are in weeks).

Normal

A.

B.

C.

D.

E.

F.

G.

H.

Activity

Solicit Volunteers

Initial “Free” Exposure

Raise Money

Organize Schedule

Hire Advertising Firm

Arrange TV Interview

Advertising Campaign

Personal Campaigning

Crash

Time

6

3

9

4

2

3

5

7

Cost

$5,000

$4,000

$4,000

$1,000

$1,500

$4,000

$7,000

$8,000

Immediate Predecessors

Time

Cost

$9,000

4

--3

$4,000

--6

$10,000

A

2

$2,000

A

1

$2,000

B

1

$8,000

B

4

$12,000

C,E

5

$20,000

D,F

Joe King is not a wealthy man and would like to organize a 16-week campaign at minimum

cost. (i) Model his problem as a project network, (ii) identify the critical path employing

CPM. Using your solution of CPM, (iii) develop a crash plan to complete the campaign in the

desired time. Please note that you might have to draw multiple project network diagrams to

update your critical path with each crashing event by week. Your answer will show the EST,

EFT, LST and LFT (EST translates to « early start time » and it works the same for the other

acronyms).

(3)

(a)

The production process of a company has a process capability ratio () of 0.78.

What is the sigma rating of the quality of the production process? (5 points)

(b)

What is the quality rating in DPMO? (5 points)

(c)

The production cost of a good product is $28; bad products are resold to recover 25%

of cost. What is the effective average cost per unit of saleable product? What should be the

price earn 20% profit on effective cost? (10 points)

(d)

During the course of production, the maintenance engineer skips routine maintenance

on two consecutive dates. Investigation reveals the process capability index to have decreased

to 0.7, with more products falling below the LTL. What fraction of products is defective

under this condition of low maintenance? (Give a breakdown of defect types—below LTL

and above UTL. What is the effective cost per unit of saleable product? What is the loss per

unit incurred because of the process drift? (10 + 5=15 points)

(e)

Consider the situation when , when the process is centered. Suppose bad products

cannot always be detected perfectly; that there is only 85% chance that a bad product will be

discovered before sale. When a bad product is discovered, it is resold as before to recover

25% of cost. A bad product if sold to a customer has a 95% chance of being reported. When

reported, you replace the bad product free, refund 100% of the price and give a voucher

worth $20 as compensation for inconvenience. What is the effective cost per unit of revenue

generating product? What should be your price to earn a 20% profit on effective cost? (15

points)

(f) (This part has no contextual relation with the foregoing problems) A manufacturer

manufactures mechanical gears with the following radial specs : cm in radius. If its

production process is known to have a standard deviation of 0.005 cm, calculate its process

capability and its sigma-rating. (5+5=10 points)

(4)

Wegafolks Groceries usually has three cashiers working on Monday afternoons. The

customers arrive at the checkout area at a Poisson rate of fifteen every half an hour, and

randomly choose one of the lines dedicated to each of the cashiers. The cashiers can checkout

a customer on the average in five minutes; the service time follows an exponential probability

distribution.

(a)

your

How often is a cashier idle (any one)? (5 points) (Specify the model you use and show

calculation)

(b)

total

What is the average number of customers waiting for each of cashiers, and average

customers waiting in the check out area as a whole? (7.5 +2.5=10points)

(c)

What is the probability that an arriving customer will not have to wait to check out?

(Hint: probability that all cashiers are busy; note that they are independent variables)

(7.5 points)

(d)

What is the average time it takes a customer to check out? (5 points)

(8)

Wegfolks expects that the customer arrival rate will increase and will be 54 per hour

after the

opening of a new processed food section. In view of this it wants to investigate the

effect of

setting up a queue structure like airports---one queue serving all the cashiers. In doing

all the

following problems assume this increased customer arrival rate for all questions that

follow

(a)

What is the minimum number of cashiers it must hire to avoid overcrowding ,

that is

avoid forming infinitely long lines?

(5 points)

For the rest of the questions assume that it has now hired FIVE cashiers and there is

only

ONE queue in the store.

(a) What would be the average waiting time for a customer? (10 points)

(b) What fraction of customers would have to get in the queue before check out? (10

points)

(c) On the average how much time must a customer budget to check out? (5 points)

(5) Consider the followind data about a simple project (note that the table gives the expected

time and the standard deviation of completion time for each activity). (a) Draw the

corresponding project network AS WE HAVE DONE IN CLASS. (b) Identify the critical

path, (c) specify expected project completion time and (d) the standard deviation of the

project completion time. (e) What is the chance that the project would be completed between

40 and 45 weeks to complete? (f) Imagine that you are a contractor and plan to place a bid for

a project contract. (g)Apart from a price offer you need to propose a project completion time

that you want to achieve with 99% certainty, what would be the minimum time you will state

in you bid ?(5+10 (b,c)+5+5+5+10)

Activity

A

B

C

D

E

F

G

H

Immediate

Predecessor(s)

----A

B

B

C,D

D,E

G

Expected

Time(Weeks)

4

5

7

10

7

8

9

10

Standard Dev. of

Time (Weeks)

3

1

3

2

1.5

2.2

4

2

(6) Mrs. Rich has just bought a new $30,000 car. As a reasonably safe driver, she believes

that there is only a 5% chance of being in an accident in the forthcoming year. If she is

involved in an accident, the damage to her new car depends on the severity of the accident.

The probability distribution for the range of possible accidents and the corresponding damage

amounts (in dollars) are shown in the table below. Mrs. Rich is trying to decide whether she

is willing to pay $170 each year for collision insurance with a $300 deductible. Note that with

this type of insurance, she pays the first $300 in damages if she causes an accident, and the

insurance company pays the remainder.

Distribution of Accident Types and Corresponding Damage Amounts

Type of Accident

Minor

Moderate

Serious

Catastrophic

Conditional Probability

0.60

0.20

0.10

0.10

Damage to Car

$200

$1,000

$4,000

$30,000

(i) Model her decsion problem as a decision tree. (8)

(ii) Identify her optimal decision for the given deductible.(12)

(iii)

Find the maximum deductible that will make her indifferent between buying the

policy and not buying the policy. (5)

(7) Consider an oil-wildcatting problem. You have mineral rights on a piece of land that you

believe may have oil underground. There is only a 10% chance that you will strike oil if

you drill, but the the payoff is $200,000. It costs $10,000 to drill. The alternative is not to

drill at all, in which case your profit is zero.

(a) Draw a decision tree to model your decision problem and determine your optimal

decision

and the value of the decsion problem. (5)

(b) Now suppose you have access to the services of the « omniscient Yoda » who blinks

his

right eye if he sees oil in your land and blinks his left eye if he doesn’t see oil in your

land.

In short, Yoda guve you perfect information before you can make your decision. What

is the

value of perfect information provided by Yoda. What use is this information to you ?

(10)

(c) Now suppose that before you drill, you might consult a geologist who can assess the

promise

of the piece of land. She can tell you whether your prospects are « good » or « poor ».

But,

unlike Yoda, she is not a perfect predictor. If there is oil there is a 95% chance that

she will

say the prospects are « good » and if there is no oil then there is only a 10% chance

that she

will say that the prospects are « good ».

(i)

consult

Draw an augmented decision tree that also included the decision of whether to

the geologist or not. (10)

(i)

What is your optimal decision (consulting or not consulting the geologist) and

the

associated action strategy ? (20)

(iii)

What is the EVSI and the efficiency of the information ? (5+5=10)

(iv)

If she charges $7000, would you hire her ? (5)

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